Kolmogorov Continuity Theorem

In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.

Statement

Let $(S,d)$ be some complete separable metric space, and let $X\colon [0, + \infty) \times \Omega \to S$ be a stochastic process. Suppose that for all times $T > 0$, there exist positive constants $\alpha, \beta, K$ such that

:$\mathbb [d(X_t, X_s)^\alpha] \leq K , t - s , ^$

for all $0 \leq s, t \leq T$. Then there exists a modification $\tilde$ of $X$ that is a continuous process, i.e. a process \tilde\colon sample-continuous;
* for every time $t \geq 0$, $\mathbb (X_t = \tilde_t) = 1.$

Furthermore, the paths of $\tilde$ are locally $\gamma$-Hölder-continuous for every $0<\gamma<\tfrac\beta\alpha$.

Example

In the case of Brownian motion">Hölder continuity">$\gamma$-Hölder-continuous for every $0<\gamma<\tfrac\beta\alpha$.

Example

In the case of Brownian motion on $\mathbb^n$, the choice of constants $\alpha = 4$, $\beta = 1$, $K = n (n + 2)$ will work in the Kolmogorov continuity theorem. Moreover, for any positive integer $m$, the constants $\alpha = 2m$, $\beta = m-1$ will work, for some positive value of $K$ that depends on $n$ and $m$.

See also

* Kolmogorov extension theorem

References

* {{cite book , author= Daniel W. Stroock, S. R. Srinivasa Varadhan , authorlink=Daniel W. Stroock, S. R. Srinivasa Varadhan , title=Multidimensional Diffusion Processes , publisher=Springer, Berlin , year=1997 , isbn=978-3-662-22201-0 p. 51

Theorems about stochastic processes